Independent group topologies on Abelian groups

Abstract

Two non-discrete T 1 topologies τ 1, τ 2 on a set X are called independent if their intersection τ 1∩ τ 2 is the cofinite topology on X. We show that a countable group does not admit a pair of independent group topologies. We use MA to construct group topologies on the additive groups R and T independent of their usual interval topologies. These topologies have necessarily to be countably compact and cannot contain convergent sequences other than trivial. It is also proved that all proper unconditionally closed subsets of an Abelian (almost) torsion-free group are finite. Finally, we generalize the result proved for R and T by showing that every second countable group topology on an Abelian group of size 2 ω without non-trivial unconditionally closed subsets admits an independent group topology (this also requires MA). In particular, this implies that under MA, every (almost) torsion-free Abelian group of size 2 ω admits a Hausdorff countably compact group topology.